Economics 201B–Second HalfLecture 1, 3/9/10Overview of General Equilibrium TheoryEdgeworth Box ModelGeneral EquilibriumStr ategicHoly Grail, 201Ω Partial Equilibrium General EquilibriumStrategic Price-Taking201A, 201B(1sthalf) 201B (2ndhalf)IO Partial EquilibriumPrice-Taking201A(1sthalf)Price-Taking: Assumes competitive market, each agent is negligible, so agents take prices as given.Benchmark against which one compares:• Distortion from Monopoly or Oligopoly: Industrial Organization• Distortion arising from Taxes: Public Finance• Effects of Welfare Payments on Labor Supply• Externalities• Most of Finance is competitive• Freshwater Macro1Features:• Very successful model of exchange (taken production decisions as given, explains how consumptionis allocated)• Less succesful model of production (should be strategic)• Method of analyzing effects of policy changes, such as trade liberalization or tax changesFocus on Proofs and Precision:• Further develop skills needed to read journal articles• Specify model completely• Know what you know• Know what you don’t know; public discourse is incomplete or misleadingProblem Sets:• Most important part of course• Work together• Write up solutions on your ownFundamental Notion: Walrasian Equilibrium, AKA Competive Equilibrium2Vector of prices and allocations of goods and production plans such that• Consumers maximize utility, taking prices as given• Firms maximize profit, taking prices as given• All markets clear (Supply = Demand)Four Fundamental Questions: (G´erard Debreu made fundamental contributions to all four 1950-1975.Theory of Value, 1959, won him a Nobel Prize, but not tenure at Yale, so he came to Berkeley)1. Existence: Arrow-Debreu 1954, Debreu 1959.• As soon as equilibrium notion is defined, must resolve the existence question for a reasonablybroad class of models as a basic consistency check.• If existence fails, qualitative results on equilibrium have no foundation.• Now, you can’t publish a model and solution concept without showing existence.• Requires Convexity of Preferences (not great assumption) and Technology (horrible assump-tion). We’ll discuss what happens without convexity.32. Welfare Theorems:• Par eto Optimality: There is no way to rearrange things to make everyone better off.• First Welfare Theorem: Every Walrasian Equilibrium is Pareto Optimal. Adam Smith–Arrow–Debreu– True in great generality– Main assumption–price-taking–is hidden in defintion of Walrasian Equilibrium– Fails with∗ Externalities∗ Incomplete Markets∗ Overlapping Generations• Second Welfare Theorem: Every Pareto Optimum is a Walrasian Equilibrium with IncomeTransfers. Debreu (1959)– Walrasian Equilibrium is not biased to certain Pareto Optima over others– Informational Efficiency: Even though government can’t compute Pareto Optima, it canachieve any Pareto Optimum by allocating income and relying on First Welfare Theoremto generate Pareto Optimality– Needs Convexity of Preferences and Technology. We’ll look at what happens if convexityfails.– Income Transfers need lump sum taxes43. Uniqueness and Determinacy: Does an economy have a unique Walrasian Equilibrium: No! Basicassumptions do not imply uniqueness. Best work in literature uses assumptions on distribution ofcharacteristics: Hildenbrand–Grandmont–Quah.• Next Best: Determinacy Debreu (1970).– For “most” economies,∗ there are only finitely many equilibria∗ the equilibria move in a smooth was as the parameters of the economy change– A foundation for comparative statics: Effect of∗ tax change∗ change in minimum wage∗ tariff reduction– Most of the central policy questions are comparative statics questions.– There are nondifferentiable approaches to monotone comparative statics (Milgrom, Shan-non, others)54. Is the price-taking assumption justified? Most prominent formulation is cor e c onver gence Edgeworth1881–Shubik 1959–Debreu-Scarf 1964; much additional work through mid-1980s.• Core is institution-free model of what outcomes could reasonably result from trade• Core convergence means roughly “Every core allocation is nearly Walrasian”End of OverviewFirst Model: Edgeworth Box (2 persons, 2 goods, exchange economy) Either• No production; or• Production decisions made exogenously, focus on the question of how consumption is allocatedFeatures:• 2consumersi =1, 2• 2 goods  =1, 2• Consumption space is R2. Each agent’s consumption set is R2+.• Not interested in minimal assumptions just yet.6• i’s endowment ωi∈ R2+; ωi: i’s endowment of good • social endowment: ¯ω = ω1+ ω2; social endowment of good :¯ω= ω1+ ω2:¯ω =(¯ω1, ¯ω2)• An allocation x ∈R2+ 2is– feasible if x1+ x2≤ ¯ω– exact if x1+ x2=¯ω. Every exact allocation is a point in the Edgeworth Box, and vice versa.Book says non-wasteful, I don’t like this because it assumes goods are goods and not bads.• Agent i is endowed with a preference relation ion R2+which is1. complete: x, y ∈ R2+⇒ (x iy ∨ y ix)2. transitive:(x iy ∧ y iz) ⇒ x iz3. strictly convex:(y ix, z ix, y = z) ⇒∀α∈(0,1)αy +(1− α)z ix4. continuous:(xniyn,xn→ x, yn→ y) ⇒ x iy5. strongly monotone: y ≥ x, y = x ⇒ y ix(x iy means x iy and y ix)7• Price p ∈ R2+• Budget Set: Bi(p)={x ∈ R2+: p · x ≤ p · ωi} Observe that Bi(p) may extend outside the EdgeworthBox.• Par eto Optimality in Edgeworth Box An exact allocation x is– Pareto Optimal if there is no other exact allocation x withx iixifor both ix iixifor some i– Weakly Pareto Optimal if there is no other exact allocation x withx iixifor both i• If preferences are smooth, interior Pareto Optima are points of tangency of indifference curves, socan be computed by equating Marginal Rates of Substitution. But observe:– If a Pareto Optimum lies on the boundary of the Edgeworth Box, tangency typically fails, andMarginal Rates of Substitution are typically not equated across agents.8– If preferences are not smooth (for example, they have kinks), then Pareto Optima need not bepoints of tangency. Indeed, Pareto Optima are more likely to be at the kink points than atpoints where the preference is smooth.• Boundary consumptions do matter. A typical person consumes zero quantity of nearly all goods.So Pareto